121616949-math.131 - 6.1 Optimization 117 area as the clear...

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6.1 Optimization 117 area as the clear glass ( k is between 0 and 1). If the distance from top to bottom (across both the rectangle and the semicircle) is a fixed distance H , find (in terms of k ) the ratio of vertical side to horizontal side of the rectangle for which the window lets through the most light. 23. You are designing a poster to contain a fixed amount A of printing (measured in square centimeters) and have margins of a centimeters at the top and bottom and b centimeters at the sides. Find the ratio of vertical dimension to horizontal dimension of the printed area on the poster if you want to minimize the amount of posterboard needed. 24. The strength of a rectangular beam is proportional to the product of its width w times the square of its depth d . Find the dimensions of the strongest beam that can be cut from a cylindrical log of radius r . . . | | d | | ←- w -→ Figure 6.6 Cutting a beam. 25. What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere?
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